Duality for a Convex Fractional Programming

International Journal of Optimization: Theory, Methods and Applications

2070-5565(Print) 2070-6839(Online) www.gip.hk/ijotma © 2009 Global Information Publisher (H.K) Co., Ltd. 2009, Vol. 1, No. 3, 291-301.

Duality for a Convex Fractional Programming under Fuzzy Environment Pankaj Gupta*, Mukesh Kumar Mehlawat Department of Operational Research, University of Delhi, Delhi-110007, India {[email protected], [email protected]}

Abstract. In this paper, we study a particular type of convex fractional programming problem and its dual under fuzzy environment. We present appropriate duality results for a fuzzy environment using aspiration level approach. This study use linear membership functions to represent fulfillment of the decision maker’s degree of satisfaction. Keywords: Fuzzy set theory, fractional programming, duality theory, primal-dual problems.

1 Introduction Mathematical programming finds an extensive use in facilitating managerial decision situations in a large number of domains. An important class of mathematical programming problems is fractional programming which deals with situations where a ratio between physical and/or economical functions, is maximized (or minimized). There are many decision situations that necessitate consideration of uncertainties in working environment best captured by fuzzy set theory. The concept of decision making in fuzzy environment was first proposed by Bellman and Zadeh [1]. Subsequently, Tanaka et al. [2] made use of this concept in mathematical programming. The use of fuzzy set theory in fractional programming has been discussed by many authors, e.g. Luhandjula [3], Dutta et al. [4], Ravi and Reddy [5], Gupta and Bhatia [6], Chakraborty and Gupta [7], Stancu-Minasian and Pop [8], Mehra et al. [9], Pop and Stancu-Minasian [10]. * Corresponding Author. Email: [email protected].

GLOBAL INFORMATION PUBLISHER

291

International Journal of Optimization: Theory, Methods and Applications

The main focus in fuzzy fractional programming like other fuzzy mathematical programming problems has been placed on how to identify the best compromise solution yielding the highest possible degree of membership(overall satisfaction). It is natural therefore that the duality theory plays an important role for attaining the goal in a fuzzy environment as well. However, only few studies exploring duality in fuzzy fractional programming have appeared in literature. Lee et al. [11] developed a parallel algorithm and studied duality for a fuzzy multiobjective linear fractional programming problem. Wu [12] developed duality theory in fuzzy optimization problems formulated by the Wolfe’s primal and dual pair. Gupta and Mehlawat [13] studied duality for fuzzy linear fractional programming under fuzzy environment using both linear and nonlinear membership functions. In this paper, we attempt to generate duality results for a particular type of convex fractional programming problem and its dual under fuzzy environment using the aspiration level approach described by Zimmermann [14]. The aspiration level approach used in the present study is based on the fact that in practice a decision maker is more comfortable describing fuzzy constraints or establishing aspiration levels for the objective and/or constraints than specifying a large number of fuzzy numbers for the various parameters of the problem. Moreover, the aspiration level approach is more effective as it does not require consistency in the decision maker’s judgment; the aspiration levels are more like a probe than weighting parameters. The present research relies on the fact that the duality between fuzzy primal-dual pair can be relaxed (with additional terms) from the conventional duality (without additional terms) owing to the presence of fuzzy constraints (soft constraints) as developed by Bector and Chandra [15] for fuzzy primal-dual linear programming problems. However, the duality results generated for a fuzzy environment must conform to the corresponding results for the crisp situations. This implies that the duality results for a fuzzy environment are a special case and lead to the corresponding results from crisp situations. The concept of duality for a fuzzy environment used in the present study is well supported by a significant amount of prior research for fuzzy primal-dual linear programming problems, e.g. Hamacher et al. [16], Rödder and Zimmermann [17], Liu et al. [18], Bector and Chandra [15, 19], Bector et al. [20, 21], Vijay et al. [22]. This paper is organized as follows. In Section 2, we consider a pair of fuzzy primal-dual fractional programming problems in which vague aspiration levels are represented by linear membership functions. We establish appropriate duality results for a fuzzy environment. In Section 3, we present numerical illustrations of the duality results and related situations. Finally, some concluding remarks are made in Section 4.

2 Duality for a Fuzzy Convex Fractional Programming Problem We consider the following convex fractional programming problem and its dual studied in [23]. (PCFPP) min f ( x) = subject to

292

GLOBAL INFORMATION PUBLISHER

(c t x ) 2 dtx

Duality for a Convex Fractional Programming under Fuzzy Environment

Ax ≥ b , x ≥ 0,

(DCFPP) max g (u, v) = bt u subject to At u + dv 2 ≤ 2cv , u, v ≥ 0 ,

where A is an m × n matrix, x , c and d are column vectors with n components, b is a column vector with m components. Let S = {x | Ax ≥ b, x ≥ 0} be the domain of feasible solutions for the primal problem (PCFPP). On domain S, assume that c t x ≥ 0 and d t x > 0. Let T = {(u, v) | u ≥ 0, v ≥ 0, At u + dv 2 ≤ 2cv} be the domain of feasible solutions for the dual problem (DCFPP). Consider following fuzzy versions of the (PCFPP) and the (DCFPP) in the sense of Zimmermann [14]. Let us call them (FPCFPP) and (FDCFPP). (FPCFPP)Find x ∈ R n subject to f ( x) = Ax > b , % x ≥ 0.

(c t x ) 2 < z0 , dt x %

(FDCFPP)Find u ∈ R m , v ∈ R subject to g (u, v) = bt u > w0 , % At u + dv 2 < 2cv , % u, v ≥ 0 .

Here “ < ” and “ > ” indicates that the inequalities are flexible and may be described by a fuzzy % % set whose membership function represents fulfillment of the decision maker’s degree of satisfaction and have interpretation of “essentially less than” and “essentially greater than” in the sense of Zimmermann [14]. Also z0 and w0 are aspiration levels of the two objectives. Further, let p0 , pi (i = 1, 2, L, m), be subjectively chosen constants of admissible violations associated with the objective function and the constraints of the problem (PCFPP). Next, we define linear membership functions μi : R → [0,1] to obtain a degree of satisfaction in the problem (FPCFPP). ⎧ 1 if f ( x) ≤ z0 , ⎪ ⎪ f ( x ) − z0 μ0 ( x) = ⎨1 − if z0 < f ( x ) ≤ z0 + p0 , p0 ⎪ ⎪ 0 if f ( x) > z0 + p0 , ⎩

(1)

GLOBAL INFORMATION PUBLISHER

293

International Journal of Optimization: Theory, Methods and Applications

Fig. 1 Membership function for the objective function value goal

and ⎧ 1 if Ai ⋅ x ≥ bi , ⎪ ⎪ b − Ai ⋅ x μi ( x) = ⎨1 − i if bi − pi ≤ Ai ⋅ x < bi , pi ⎪ ⎪ 0 if Ai ⋅ x < bi − pi . ⎩

(2)

Fig. 2 Membership function for the fuzzy inequality constraint Ai ⋅ x − bi > 0 %

Using the “min” operator to aggregate the overall satisfaction and following Zimmermann [14], the crisp equivalent of the problem (FPCFPP) is

294

GLOBAL INFORMATION PUBLISHER

Duality for a Convex Fractional Programming under Fuzzy Environment

(CP) min − λ subject to (λ − 1) p0 ≤ z0 − f ( x) , (λ − 1) pi ≤ Ai ⋅ x − bi ,

(3) (i = 1, 2,L , m) ,

(4)

λ ≤ 1,

(5)

x, λ ≥ 0 ,

(6)

where Ai ⋅ (i = 1, 2,L , m) denotes the i th row of the matrix A and bi is the i th component of b. Let q j ( j = 0,1, 2, L, n) be subjectively chosen constants of admissible violations of the objective and the constraint functions of the problem (DCFPP). On the same lines as above, we obtain crisp equivalent of the problem (FDCFPP) as (CD) max η subject to (η − 1)q0 ≤ g (u, v) − w0 ,

(7)

(η − 1)q j ≤ 2c j v − Atj ⋅ u − d j v 2 , ( j = 1, 2, L, n),

(8)

η ≤ 1,

(9)

u, v,η ≥ 0 .

(10)

Next, we establish appropriate duality results under fuzzy environment for the pair (CP), (CD). Theorem 1. Let ( x, λ ) be feasible for (CP) and (u, v,η ) be feasible for (CD). Then

(η − 1)q t x + (λ − 1) p t u ≤ f ( x) − g (u, v) .

(11)

Proof: Assume that S ≠ φ and T ≠ φ . Multiply (8) by x ≥ 0, we get

(η − 1)q t x ≤ 2 xt cv − xt At u − xt dv 2 .

(12)

Also, multiply (4) by u ≥ 0, we get

(λ − 1) p t u ≤ xt At u − bt u .

(13)

Adding (12) and (13), we get

(η − 1)q t x + (λ − 1) p t u ≤ 2 xt cv − xt dv 2 − bt u , or (η − 1)q t x + (λ − 1) p t u ≤ −bt u − (v(d t x)1/ 2 − (c t x)( d t x) −1/ 2 ) 2 +

(c t x ) 2 , dtx

GLOBAL INFORMATION PUBLISHER

295

International Journal of Optimization: Theory, Methods and Applications

or (η − 1)q t x + (λ − 1) p t u + bt u + (v(d t x)1/ 2 − (c t x)(d t x) −1/ 2 ) 2 −

(c t x ) 2 ≤ 0. dt x

Therefore, (η − 1)q t x + (λ − 1) p t u + bt u −

(c t x ) 2 ≤ 0, dtx

or (η − 1)q t x + (λ − 1) p t u ≤ f ( x) − g (u, v) .

Hence the result. The inequality (11) as defined above is a generalization of the crisp weak duality result, i.e. the crisp weak duality inequality g (u, v) ≤ f ( x) is generalized to the fuzzy weak duality inequality g (u, v) < f ( x). In an economic analysis of the weak duality result, the fuzzy inequality assumes that the % decision maker is able to maximize or minimize the “utility functions” and thus take into consideration all possible decisions of the competitors. Then, the decision maker can draw upon preference functions for his losses from a simultaneous consideration of the competitor’s decision and his own. Thus, we obtain a special kind of bounded relationship between the solutions that satisfy the decision maker’s assessments of prices and quantities in competitive situations, as opposed to a strict optimizer such as a maximizer or minimizer. In order to satisfy the fuzzy inequality, some tolerance must be given for the satisfying solutions of the decision makers, which is, in general, not bounded. By relaxing the fuzzy inequality, we allow the decision makers to make decisions that maximize (or minimize) their utility functions, subject to the bounded relationship defined in the inequality (11). Remark 1. It may be noted that for λ = 1 and η = 1 (i.e. when the original problems are crisp), the inequality (11) reduces to g (u, v) ≤ f ( x) , which is the standard weak duality result in the crisp duality theory. Also, for 0 < λ < 1 and 0 < η < 1, the situation remains fuzzy. For given tolerance levels ( p = p1 , p2 ,L, pm ) and (q = q1 , q2 ,L, qn ), the situation is quantified in the following expression:

(η − 1)q t x + (λ − 1) p t u Remark 2. In addition to inequality (11), adding (3) and (7), we can arrive at (λ − 1) p0 + (η − 1)q0 ≤ ( g (u, v ) − f ( x)) + ( z0 − w0 ).

(14)

It may also be noted that the inequality (14) relates the relative difference of the aspiration levels z0 of f ( x), and w0 of g (u, v ), respectively, in terms of their tolerance levels p0 and q0 . If in addition to λ = 1 and η = 1 , we also have z0 − w0 = 0, then combining the inequalities (11) and (14) gives f ( x) = g (u, v) , i.e. x and w are optimal for the crisp problems (PCFPP) and (DCFPP). Since (CP) and (CD) are not dual in the conventional sense but only crisp equivalents of the fuzzy pair (FPCFPP) and (FDCFPP), there may not be a direct or converse duality between them. How-

296

GLOBAL INFORMATION PUBLISHER

Duality for a Convex Fractional Programming under Fuzzy Environment

ever, like the crisp convex fractional programming duality scenario, we can verify the extent to which the crisp equivalent (11) of the fuzzy weak duality inequality is satisfied as an equality. Theorem 2. Let ( x , λ ) be feasible for (CP), and let (u , v ,η ) be feasible for (CD) such that (i) (η − 1)q t x + (λ − 1) p t u = f ( x ) − g (u , v ) (ii) (η − 1)q0 + (λ − 1) p0 = ( g (u , v ) − f ( x )) + ( z0 − w0 ) (iii) The aspiration levels z0 and w0 satisfy w0 − z0 ≤ 0 then ( x , λ ) is optimal to (CP) and (u , v ,η ) is optimal to (CD). Proof: Let ( x, λ ) be feasible for (CP) and (u, v,η ) be feasible for (CD). Using theorem 1, we have (η − 1)q t x + (λ − 1) p t u − ( f ( x) − g (u, v)) ≤ 0 .

(15)

From (i) we are given that (η − 1)q t x + (λ − 1) p t u = f ( x ) − g (u , v ) .

(16)

The relations (15) and (16) imply that for any feasible solution ( x, λ ) of (CP) and for any feasible solution (u, v,η ) of (CD), we have (η − 1)q t x + (λ − 1) p t u − ( f ( x) − g (u, v)) ≤ (η − 1)q t x + (λ − 1) p t u − ( f ( x ) − g (u , v )).

This further implies that ( x , λ , u , v ,η ) is optimal to the following problem whose maximum value is zero, max ( (η − 1)q t x + (λ − 1) p t u − ( f ( x) − g (u, v)) ) subject to (λ − 1) p0 ≤ z0 − f ( x) , (λ − 1) pi ≤ Ai ⋅ x − bi , (i = 1, 2, L, m), (η − 1)q0 ≤ g (u, v) − w0 , (η − 1)q j ≤ 2c j v − Atj ⋅ u − d j v 2 , ( j = 1, 2, L, n),

λ ,η ≤ 1, x, u, v ≥ 0 , λ ,η ≥ 0 .

Now from the given condition (i), we have

(η − 1)q t x + (λ − 1) p t u − ( f ( x ) − g (u , v )) = 0 .

(17)

Also from given condition (ii), we have

(η − 1)q0 + (λ − 1) p0 − ( g (u , v ) − f ( x )) − ( z0 − w0 ) = 0 .

(18)

Adding (17) and (18), we get (η − 1)q t x + (λ − 1) p t u + (λ − 1) p0 + (η − 1)q0 + ( w0 − z0 ) = 0 .

GLOBAL INFORMATION PUBLISHER

297

International Journal of Optimization: Theory, Methods and Applications

Since each term in the above sum is non-positive (because λ ,η ≤ 1 ) and, therefore, each of these terms should each be equal to zero, i.e. (η − 1)q t x = 0 , (λ − 1) p t u = 0 , (λ − 1) p0 = 0, (η − 1)q0 = 0, w0 − z0 = 0 .

Since (λ − 1) p0 ≤ 0 and (η − 1) q0 ≤ 0 (because λ ,η ≤ 1 ), we get (λ − 1) p0 ≤ (λ − 1) p0 ,

and (η − 1)q0 ≤ (η − 1)q0 .

But p0 > 0 and q0 > 0, so canceling p0 and q0 , we get −λ ≥ −λ (or λ ≤ λ ) and η ≤ η . Hence the result. In the present paper, since we are studying duality in a fuzzy environment, even if there is a strong duality between the fuzzy primal-dual pair (CP) and (CD) in the sense of Theorem 2, strong duality will be achieved only in the fuzzy sense as g (u, v ) −% f ( x). Remark 3. In crisp situations (i.e. when λ = 1 and η = 1 ), we get the following expression: (η − 1)q t x + (λ − 1) p t u = 0. Following (i), (ii), and (iii) of Theorem 2, we obtain g (u, v) = f ( x ), i.e. strong duality in the crisp sense.

3 Numerical Illustration Consider the following pair of primal-dual fractional programming problems. (PCFPP) min f ( x) =

(2 x1 + x2 ) 2 x1 + 2 x2

subject to 2 x1 + x2 ≥ 6 , x1 + 3x2 ≥ 8 , x1 , x2 ≥ 0 , (DCFPP) max g (u, v) = 6u1 + 8u2 subject to 2u1 + u2 + v 2 − 4v ≤ 0 , u1 + 3u2 + 2v 2 − 2v ≤ 0 , u1 , u2 , v ≥ 0 .

298

GLOBAL INFORMATION PUBLISHER

Duality for a Convex Fractional Programming under Fuzzy Environment

Take p0 = 2, p1 = 1, p2 = 2, and z0 = 1 for (PCFPP), the corresponding problem (CP) is min − λ subject to 2λ x1 + 4λ x2 − 3 x1 − 6 x2 + 4 x12 + x22 + 4 x1 x2 ≤ 0 , −2 x1 − x2 + λ ≤ −5 , − x1 − 3x2 + 2λ ≤ −6 ,

λ ≤ 1, x1 , x2 , λ ≥ 0 .

The optimal solution of (CP) is at x1∗ = 0, x2∗ = 5.2, λ ∗ = 0.20 and the optimal value of (CP) is −λ ∗ = −0.20. Now taking q0 = 1, q1 = 1, q2 = 2 and w0 = 1 for (DCFPP), the corresponding problem (CD) becomes max η subject to −6u1 − 8u2 + η ≤ 0 , 2u1 + u2 + v 2 − 4v + η ≤ 1, u1 + 3u2 + 2v 2 − 2v + 2η ≤ 2 ,

η ≤1 u1 , u2 , v,η ≥ 0 .

The optimal solution of (CD) is at η ∗ = 1, u1∗ = 0.40, u2∗ = 0, v∗ = 0.72 and the optimal value of (CD) is η ∗ = 1. For these optimal solutions, both the inequalities (11) and (14) are satisfied. Since we are using an aspiration level approach in the present study, it is unreasonable for a decision maker to choose aspiration levels that are too aggressive or too weak. If the aspiration levels are too aggressive, then there are no achievable solutions and the decision maker is asked to relax the aspiration levels. On the other hand, if the aspiration levels are too weak, then too many (perhaps all) solutions may be identified as feasible. Hence, when using the satisfying approach, the decision maker should choose his aspiration levels tightly. The problems (CP) and (CD) are nonlinear programming problems. Although for medium or large-sized problems one should expect that solving these nonlinear programming problems could be computationally difficult, this is not the case. Many excellent software systems for solving nonlinear programs are available. Some of them are based on Generalized Reduced Gradient Method developed by Lasdon and Waren [24], GRG2 [25], which can handle very efficiently large scale problems. We have used LINGO [26] that is an interactive interface to GRG2 to solve problems (CP) and (CD).

GLOBAL INFORMATION PUBLISHER

299

International Journal of Optimization: Theory, Methods and Applications

4 Conclusions We limited our construction of primal-dual problems under fuzzy environment using linear membership functions in the present paper. Different crisp equivalents of the fuzzy primal-dual problems can be obtained using other types of membership functions on the basis of the decision maker’s preferences. It would be interesting to explore the possibility of establishing duality results under fuzzy environment for such pairs of fuzzy primal-dual problems. The results obtained under fuzzy environment for the dual examined in this paper may not be tenable for other types of dual for the fractional programming problem under consideration. Thus, the choice of dual plays a significant role in the theory development and hence is a major contribution in the present study. The crisp equivalents obtained in the present paper are nonlinear (even nonconvex) problems, where non linearity exists in the constraints. Software LINGO [26] has been used to solve the numerical illustrations. The crisp equivalents can also be solved using fuzzy decisive set method [27] and the modified subgradient method [28].

Acknowledgements The first author acknowledges the research grant received under a scheme for strengthening R & D Doctoral Research Programme of University of Delhi, Delhi, India.

References 1. Bellman, R. E., Zadeh, L. A.: Decision Making in a Fuzzy Environment, Management Science 17 (1970) 141-164. 2. Tanaka, H., Okuda, T., Asai, K.: On Fuzzy Mathematical Programming, Journal of Cybernetics 3 (1984) 37-46. 3. Luhandjula, M. K.: Fuzzy Approaches for Multiple Objective Linear Fractional Optimization, Fuzzy Sets and Systems 13 (1984) 11-23. 4. Dutta, D., Tiwari, R. N., Rao, J. R.: Multiple Objective Linear Fractional Programming: A Fuzzy Set Theoretic Approach, Fuzzy Sets and Systems 52 (1992) 39-45. 5. Ravi, V., Reddy, P. J.: Fuzzy Linear Fractional Goal Programming Applied to Refinery Operations Planning, Fuzzy Sets and Systems 96 (1998) 173-182. 6. Gupta, P., Bhatia, D.: Sensitivity Analysis in Fuzzy Multiobjective Linear Fractional Programming Problem, Fuzzy Sets and Systems 122 (2001) 229-236. 7. Chakraborty, M., Gupta, S.: Fuzzy Mathematical Programming for Multi-objective Linear Fractional Programming Problem, Fuzzy Sets and Systems 125 (2002) 335-342. 8. Stancu-Minasian, I. M., Pop, B.: On a Fuzzy Set Approach to Solving Multiple Objective Linear Fractional Programming Problem, Fuzzy Sets and Systems 134 (2003) 397-405.

300

GLOBAL INFORMATION PUBLISHER

Duality for a Convex Fractional Programming under Fuzzy Environment

9. Mehra, A., Chandra, S., Bector, C. R.: Acceptable Optimality in Linear Fractional Programming with Fuzzy Coefficients, Fuzzy Optimization and Decision Making 6 (2007) 5-16. 10. Pop, B., Stancu-Minasian, I. M.: A Method of Solving Fully Fuzzified Linear Fractional Programming Problem, Journal of Applied Mathematics and Computing 27 (2008) 227-242. 11. Lee, Bum-Il., Chung, Nam-Kee., Tcha, Dong-Wan.: A Parallel Algorithm and Duality for a Fuzzy Multiobjective Linear Fractional Programming Problem, Computers and Industrial Engineering 20 (1991) 367-372. 12. Wu, H. C.: Duality Theory in Fuzzy Optimization Problems Formulated by the Wolfe's Primal and Dual Pair, Fuzzy Optimization and Decision Making 6 (2007) 179-198. 13. Gupta, P., Mehlawat, M. K.: Duality for Linear Fractional Programming Problem under Fuzzy Environment, Communicated.

14. Zimmermann, H. -J.: Fuzzy Set Theory and Its Applications, 4th ed., Kluwer Academic Publishers, Dordercht (2001). 15. Bector, C. R., Chandra, S.: On Duality in Linear Programming under Fuzzy Environment, Fuzzy Sets and Systems 125 (2002) 317-325. 16. Hamacher, H., Leberling, H., Zimmermann, H. -J.: Sensitivity Analysis in Fuzzy Linear Programming, Fuzzy Sets and Systems 1 (1978) 269-281. 17. Rödder, W., Zimmermann, H. -J.: Duality in Fuzzy Linear Programming, in: A. V. Fiacco, K. O. Kortanek, (Eds.), Extermal Methods and System Analysis (1980) 415-429, Berlin, New York. 18. Liu, Y. J., Shi, Y., Liu, Y. H.: Duality of Fuzzy MC2 Linear Programming: A Constructive Approach, Journal of Mathematical Analysis and Applications 194 (1995) 389-413. 19. Bector, C. R., Chandra, S.: Fuzzy Mathematical Programming and Fuzzy Matrix Games, Berlin Heidelberg, Springer-Verlag. (2005) 20. Bector, C. R., Chandra, S., Vijay, V.: Matrix Games with Fuzzy Goals and Fuzzy Linear Programming Duality, Fuzzy Optimization and Decision Making 3 (2004) 255-269. 21. Bector, C. R., Chandra, S., Vijay, V.: Duality in Linear Programming with Fuzzy Parameters and Matrix Games with Fuzzy Pay-offs, Fuzzy Sets and Systems 146 (2004) 253-269. 22. Vijay, V., Chandra, S., Bector, C. R.: Matrix Games with Fuzzy Goals and Fuzzy Payoffs, Omega 33 (2005) 425-429. 23. Stancu-Minasian, I. M: Fractional Programming: Theory, Methods and Applications, Dordrecht, Kluwer Academic Publishers. (1997) 24. Lasdon, L., Waren, A.: Design and Testing of a Generalized Relaxed Gradient Code for Nonlinear Programming, ACM Transactions on Mathematical Software 4 (1978) 34-50. 25. Smith, S., Lasdon, L.: Solving Large Sparse Nonlinear Programs using GRG, ORSA Journal on Computing 4 (1992) 3-15. 26. Scharge, L.: Optimization Modeling with LINDO, Duxbury Press, CA. (1997) 27. Sakawa, M., Yana, H.: Interactive Decision Making for Multi-objective Linear Fractional Programming Problems with Fuzzy Parameters, Cybernetics Systems 16 (1985) 377-397. 28. Gasimov, R. N.: Augmented Lagrangian Duality and Nondifferentiable Optimization Methods in Nonconvex Programming, Journal of Global Optimization 24 (2002) 187-203.

GLOBAL INFORMATION PUBLISHER

301